Towards a wave-extraction method for numerical relativity. V. Extracting the Weyl scalars in the quasi-Kinnersley tetrad from spatial data

TL;DR

This paper develops a method to extract Weyl scalars in the quasi-Kinnersley tetrad from spatial data, fixing gauge ambiguities to analyze gravitational wave polarization in numerical relativity.

Contribution

It introduces a gauge-fixing procedure to determine Weyl scalars and their phases from spatial data, enhancing gravitational wave extraction techniques in numerical relativity.

Findings

Successfully extracts Weyl scalars from initial data sets.

Provides explicit perturbative calculations of Weyl scalars.

Clarifies gauge fixing for purely electric Weyl curvature cases.

Abstract

We extract the Weyl scalars $\Psi_0$ and $\Psi_4$ in the quasi-Kinnersley tetrad by finding initially the (gauge--, tetrad--, and background--independent) transverse quasi-Kinnersley frame. This step still leaves two undetermined degrees of freedom: the ratio $|\Psi_0|/|\Psi_4|$, and one of the phases (the product $|\Psi_0|\cdot |\Psi_4|$ and the {\em sum} of the phases are determined by the so-called BB radiation scalar). The residual symmetry ("spin/boost") can be removed by gauge fixing of spin coefficients in two steps: First, we break the boost symmetry by requiring that $\rho$ corresponds to a global constant mass parameter that equals the ADM mass (or, equivalently in perturbation theory, that $\rho$ or $\mu$ equal their values in the no-radiation limits), thus determining the two moduli of the Weyl scalars $|\Psi_0|, |\Psi_4|$, while leaving their phases as yet undetermined. Second, we break the spin symmetry by requiring that the ratio $\pi/\tau$ gives the expected polarization state for the gravitational waves, thus determining the phases. Our method of gauge fixing--specifically its second step--is appropriate for cases for which the Weyl curvature is purely electric. Applying this method to Misner and Brill--Lindquist data, we explicitly find the Weyl scalars $\Psi_0$ and $\Psi_4$ perturbatively in the quasi-Kinnersley tetrad.